From Eqs. (REF ) and (REF ), one can see that on the one hand, with the proper choices of the parameters \(\Lambda _\gamma \)  and \(\Lambda _l\) , the positive \(\Delta a_\mu ^{\text{nl}}\)  and negative \(\Delta a_e^{\text{nl}}\)  can be obtained, where \(\Delta a_l^{\text{nl}}\)  is the difference between nonlocal QED and standard model. On the other hand, at leading order, the discrepancy from the SM is proportional to the square of lepton mass \(m_l\) . It is very tempting to speculate about a simultaneous new-physics origin of the results with the same mechanism to explain the experimental discrepancies for both muon and electron. As pointed in Refs. [1]}, [2]}, although a common explanation has been shown to be possible, the model building task has proved non-trivial due to the opposite sign and the scale \(\Delta a_\mu /\Delta a_e \sim m_\mu ^2/m_e^2\) . Here with the nonlocal QED, the \(g-2\)  anomalies of electron and muon can be naturally explained. The specific parameters will be determined in the next section. Certainly, when \(\Lambda _\gamma \rightarrow \infty \)  and \(\Lambda _l \rightarrow \infty \) , \(a_l\)  will change back to the local result \(\frac{\alpha }{2\pi }\) .
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